Foreign Currency Model help

[Dibbs]

Hey guys.

I am running a currency model here and I'm trying to compare my MC results with analytical formulas. The problem is that I am having a hard time finding the analytical variance.

The model is as follows :

USD follows :
r_usd(t) = x(t) + phi_usd(t) , where dx(t) = -alpha*x(t)*dt + sigma_usd*dWt

CAD follows:
r_cad(t) = y(t) + phi_cad(t), where dy(t) = -beta*y(t)*dt + sigma_cad*dZt

I am easily able to find the solutions to both r_usd(t) and r_cad(t).

Now, the currency model : df(t) / f(t) = (r_usd(t) - r_cad(t))*dt + sigma_fx*dVt

There exist no correlation between my three wiener processes for now.

My guess is that :

by integrating both sides of my currency model from 0 to T.
ln(f(T)) - ln(f(0)) = [ r_usd(t) - r_cad(t) ] | 0 to T + int(sigma_fx*dVt) from 0 to T

Then I can find my properties. But, I'm not falling back on the results from my MC simulation.

Any thoughts ?

 

[Tsotne]

Assuming nonexistence of correlation is not correct even for simplification. Not the reason for now though

 

[Dibbs]

I know it's not correct to assume that. I'm just struggling with the calculus.